Parallel and Perpendicular Lines

Points, Lines, and Planes

3.3 Parallel and Perpendicular Lines

Algebra Extension

Give students the series of problems below to begin to explore the concepts of parallel and perpendicular lines in the coordinate plane.

1. Write the equations of two lines parallel toy=3.

2. Write the equations of two lines perpendicular toy=5.

3. What is the relationship between the two lines you found for number 27?

4. Plot the points A(2,−5),B(−3,1),C(0,4),D(−5,10). Draw the lines ←→

AB and ←→

CD. What are the slopes of these lines? What is the geometric relationship between these lines?

5. Plot the pointsA(2,1),B(7,−2),C(2,−2),D(5,3). Draw the lines←→

AB and←→

CD. What are the slopes of these lines? What is the geometric relationship between these lines?

6. Based on what you discovered in numbers 4 and 5, can you make a conjecture about the slopes of parallel and perpendicular lines?

Answers

1. Any two equations in the formy=b, wherebis a constant.

2. Any two equations in the formx=b, wherebis a constant.

3. These two lines are parallel to each other.

4. slope of←→

ABequals slope of←→

CD=−⁶₅; these lines are parallel 5. slope of←→

AB=−⁵₃, slope of←→

CD=³₅; these lines are perpendicular

6. It appears that the slopes of parallel lines are the same and the slopes of perpendicular lines are opposite reciprocals.

Properties of Parallel Lines

Connections to Map Reading

Give each student a copy of the Tube map of London below. They will be using the map to find an example of each of the postulates/theorems:

1. Corresponding Angles Postulate 2. Alternate Interior Angles Theorem

3. Alternative Exterior Angles Theorem 4. Same Side Interior Angles Theorem

Students will need to prove that each of their examplesis accurate. Students may need to use a protractor. Students may also want to work in pairs.

As an extension, have students research other subway lines, such as the “L” in Chicago, the Metro in Washington DC and the Subway in New York City.

Know What? Extension

When looking at the street map of Washington DC, point out to students that all the “state” streets are transversals throughout the city. Ask students why they think these streets do not follow the grid layout. (Lettered streets run east to west, numbered streets run north to south.)

Algebra Challenge Find the values ofxandy.

1.

2.

Answers

1. x=15^◦,y=21^◦ 2. x=37^◦,y=28^◦ Proof Challenge

Use the picture to the right to complete each proof.

1. Given:lkm,skt Prove:⁶ 4∼=⁶ 10 2. Given:lkm,skt Prove:⁶ 2∼=⁶ 15 3. Given:lkm,skt

Prove:⁶ 4 and⁶ 9 are supplementary Answers

1.

T

ABLE

3.3:

Statement Reason

1.lkm,skt Given

2.⁶ 4∼=⁶ 12 Corresponding Angles Postulate

3.⁶ 12∼=⁶ 10 Corresponding Angles Postulate

4.⁶ 4∼=⁶ 10 Transitive PoC

2.

T

ABLE

3.4:

Statement Reason

1.lkm,skt Given

2.⁶ 2∼=⁶ 13 Alternate Exterior Angles Theorem

3.⁶ 13∼=⁶ 15 Corresponding Angles Postulate

4.⁶ 2∼=⁶ 15 Transitive PoC

3.

T

ABLE

3.5:

Statement Reason

1.lkm,skt Given

2.⁶ 6∼=⁶ 9 Alternate Interior Angles Theorem

3.⁶ 4∼=⁶ 7 Vertical Angles Theorem

4.⁶ 6 and⁶ 7 are supplementary Same Side Interior Angles

5.⁶ 9 an⁶ 4 are supplementary Same Angle Supplements Theorem

Proving Lines Parallel

Construction Extension

Extend Investigation 3-5 to the Converses of the Alternate Interior Angles Theorem and the Alternate Exterior Angles Theorem. You can decide to do this as a student or teacher-led activity. The major difference will be that in Step 3, the copied angle will be in a different location. Ask students how they can extend this investigation to the Converse of the Same Side Interior Angle Theorem.

Construction Challenge

Draw a straight line. Construct a line perpendicular to this line through a point on the line. Now, construct a perpendicular line to this new line. What can you conclude about the original line and this final line? How could you prove this conjecture?

Answers

Construction, the first and last lines are parallel. You might conjecture that two lines perpendicular to the same line are parallel to each other. You could prove this using any of the converse theorems learned in this section because all four angles formed where the transversal intersects the two parallel lines are right angles. Thus, Alternate Interior Angles, Alternate Exterior Angles and Corresponding Angles are all congruent and the Same Side Interior Angles are supplementary.

Connections to Map Reading

The following link is a map of the mall in Washington DC:http://www.destination360.com/north-america/us/washi ngton-dc/images/smithsonian-institution-map.jpg (found on http://www.destination360.com/north-america/us/washington- dc/smithsonian-institution

There are several different examples of parallel lines and transversals in this map. Have students to write a series of directions to take someone on a tour of the mall. They must go to the American History Museum, the National Archives, the Newseum, the West National Gallery of Art, and the Air and Space Museum. Students can pick the order in which they go to these sites. Working in pairs, their job is to write down directions with the shortest path.

Then, they need to determine if there is another set of directions that is close to (or the same length) as their original directions. When finished, have the students swap directions with a neighboring group and check to be sure that the directions work.

Properties of Perpendicular Lines

Connections to Gymnastics

Gymnasts use the parallel bars and the uneven bars in routines. Ask students if they see any perpendicular transver- sals for these pieces of equipment. Extend this discussion to other gymnastic equipment (pommel horse, rings, balance beam, vault). How would the events be different if this equipment was not parallel to the ground?

Judges also score gymnasts on their ability to get their body perfectly vertical and legs perpendicular or vertical. Are there any perpendicular transversals or parallel lines between the gymnast and the equipment or the ground?

Challenge

Findxandy. Remember that the angles in a triangle add up to 180^◦.

Answers

x=49^◦,y=111^◦

Parallel and Perpendicular Lines in the Coordinate Plane

Connections to Road Construction

Typically, the slope of a steep road, also called a grade, is measured in a percentage. If a road has a grade of 5%, which means in 100 horizontal feet the road will rise vertically 5 feet.

This would be a slope triangle for this particular piece of road. The slope would be ₁₀₀⁵ =₂₀¹. Present students with the following problem: The Grapevine on Highway 5 rises from 400 feet to 4144 feet over 6 miles. What is the grade of this road? (This is for the southbound side)

Answer

First, convert 6 miles to feet, 6×5280=31,680 f t. Now, write a ratio and then turn that into a percent. ^4144−400₃₁₆₈₀ =

3744

31680≈0.118×100%=11.8% grade. In general, if a grade is over 10% it is considered steep.

As an extension, students can find the grade of the California Incline in the Know What? at the beginning of this lesson. (11.6%)

More Connections to Construction

Slope, also called grade or pitch, is a very important part in roof design, wheelchair ramp design, and skateboard ramp design. Have students research any or all of these online to see why slope is important to these items.

Challenge

1. A line passes through (5, -2) and (-4, 1). Find the equations of the lines that are perpendicular to this one, through each point. How do these two lines relate to each other?

Answer

1. The equation of the line isy=−¹₃x−¹₃. The perpendicular line through (5, -2) isy=3x−17 and through (-4, 1) isy=3x+13. These two lines are parallel to each other.

The Distance Formula

Extension: Shortest Distance between a Point and a Line

We know that the shortest distance between two points is a straight line. This distance can be calculated by using the distance formula. Let’s extend this concept to the shortest distance between a point and a line.

Just by looking at a few line segments fromAto linel, we can tell that the shortest distance between a point and a line is theperpendicular linebetween them. Therefore,ADis the shortest distance betweenAand linel.

Example:Determine the shortest distance between the point (1, 5) and the liney=¹₃x−2.

Solution: First, graph the line and point. Second determine the equation of the perpendicular line. The opposite sign and reciprocal of ¹₃ is -3, so that is the slope. We know the line must go through the given point, (1, 5), so use that to find they−intercept.

y=−3x+b

5=−3(1) +b The equation of the line isy=−3x+8.

8=b

Next, we need to find the point of intersection of these two lines. By graphing them on the same axes, we can see that the point of intersection is (3, -1), the green point.

Finally, plug (1, 5) and (3,-1) into the distance formula to find the shortest distance.

d= q

(3−1)²+ (−1−5)²

= q

(2)²+ (−6)²

= √ 2+36

= √

38≈6.16units

Extension: Shortest Distance between Parallel Lines (slopes other than 1 or -1)

In the text, we went over how to find the distance between two parallel lines where the slope was either 1 or -1. Here, we extend this concept to any two parallel lines. It is roughly the same process, but there is substantially more math involved. This extension would probably be done best in pairs or in groups.

Example:What is the shortest distance betweeny=2x+4 andy=2x−1?

Solution: Graph the two lines and determine the perpendicular slope, which is −¹₂. Find a point ony=2x+4, let’s say (-1, 2). From here, use the slope of the perpendicular line to find the corresponding point ony=2x−1. If you move down 1 from 2 and over to the right 2 from -1, you will hity=2x−1 at (1, 1). Use these two points to determine the distance between the two lines.

d= q

(1+1)²+ (1−2)²

= q

2²+ (−1)²

= √ 4+1

= √

5≈2.24units The lines are about 2.24 units apart.

Notice that you could have used any two points, as long as they are on the same perpendicular line. For example, you could have also used (-3, -2) and (-1, -3) and you still would have gotten the same answer.

d= q

(−1+3)²+ (−3+2)²

= q

2²+ (−1)²

= √ 4+1

=

√

5≈2.24units Example:Find the distance between the two parallel lines below.

Solution:First you need to find the slope of the two lines. Because they are parallel, they are the same slope, so if you find the slope of one, you have the slope of both.

Start at they−intercept of the top line, 7. From there, you would go down 1 and over 3 to reach the line again.

Therefore the slope is−¹₃ and the perpendicular slope would be 3.

Next, find two points on the lines. Let’s use they−intercept of the bottom line, (0, -3). Then, rise 3 and go over 1 until your reach the second line. Doing this three times, you would hit the top line at (3, 6). Use these two points in the distance formula to find how far apart the lines are.

d= q

(0−3)²+ (−3−6)²

= q

(−3)²+ (−9)²

= √ 9+81

= √

90≈9.49units Extension Problems

Determine the shortest distance between the given line and point. Round your answers to the nearest hundredth.

1.y=¹₃x+4;(5,−1) 2.y=2x−4; (−7,−3) 3.y=−4x+1; (4,2) 4.y=−²₃x−8; (7,9)

Determine the shortest distance between the each pair of parallel lines. Round your answer to the nearest hundredth.

5.y=−¹₃x+2,y=−¹₃x−8 6.y=4x+9,y=4x−8 7.y=¹₂x,y=¹₂x−5

Answers 1. 6.32 units 2. 6.71 units 3. 12 units 4. 7 units 5. 9.49 units 6. 4.12 units 7. 4.47 units